Hello Garrett,
After several hours checking this, it seems like when the POST request is made, the web server is redirecting to the front end of your site and it doesn't seem to be going to Joomla's /administrator/index.php at all.
This actually leads me to believe that the web server is already redirecting the request before it even reaches Joomla. Could you try contacting your hosting provider and ask them if there are any mod_security or security module for apache that is installed which could be performing such redirection?